Learning based numerical methods for Helmholtz equation with high frequency

TL;DR

This paper introduces a learning-based numerical method for high-frequency Helmholtz equations, leveraging Tikhonov regularization and fundamental solutions to achieve accurate, efficient solutions with proven error estimates.

Contribution

It proposes a novel learning-based approach using Tikhonov regularization to stably learn the solution operator for high-frequency Helmholtz equations, enhancing efficiency and accuracy.

Findings

The method achieves high-precision solutions.

Numerical results validate the error estimates.

The approach demonstrates high efficiency and generalizability.

Abstract

High-frequency issues have been remarkably challenges in numerical methods for partial differential equations. In this paper, a learning based numerical method (LbNM) is proposed for Helmholtz equation with high frequency. The main novelty is using Tikhonov regularization method to stably learn the solution operator by utilizing relevant information especially the fundamental solutions. Then applying the solution operator to a new boundary input could quickly update the solution. Based on the method of fundamental solutions and the quantitative Runge approximation, we give the error estimate. This indicates interpretability and generalizability of the present method. Numerical results validates the error analysis and demonstrates the high-precision and high-efficiency features.